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Asymptotic behaviour near finite-time extinction for the fast diffusion equation. (English) Zbl 0885.35058

Summary: We study the Cauchy problem for the fast diffusion equation \[ u_t= \Delta(u^m),\quad u\leq 0\quad\text{in }\mathbb{R}^N\times \mathbb{R}_+, \] when \(N\geq 3\) and \(0<m<(N- 2)/N\). For a class of nonnegative radially symmetric finite-mass solutions, which vanish identically at a given time \(T>0\), we show that their asymptotic behaviour as \(t\nearrow T\) is described by a uniquely determined self-similar solution of the second kind: \(u_*(r, t)= (T-t)^\alpha f(\eta)\), where \(\eta= r/(T- t)^\beta\) and \(\alpha= (1-2\beta)/(1-m)\), and \(r=|x|\). Here \(\beta\) is determined from a nonlinear eigenvalue problem involving an ordinary differential equation for the function \(f\). Special attention is paid to the case when \(m=(N-2)/(N+ 2)\). Then \(\beta=0\) and the function \(f\) can be found explicity. The proof is based on a geometric Lyapunov-type argument and comparison arguments based on the intersection properties of the solution graphs.

MSC:

35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
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